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Classical multivariate Hermite coordinate interpolation on n-dimensional grids
In this work, we study the Hermite interpolation on n-dimensional non-equally
spaced, rectilinear grids over a field k of characteristic zero, given the
values of the function at each point of the grid and the partial derivatives up
to a maximum degree. First, we prove the uniqueness of the interpolating
polynomial, and we further obtain a compact closed form that uses a single
summation, irrespective of the dimensionality, which is algebraically simpler
than the only alternative closed form for the n-dimensional classical Hermite
interpolation [1]. We provide the remainder of the interpolation in integral
form; moreover, we derive the ideal of the interpolation and express the
interpolation remainder using only polynomial divisions, in the case of
interpolating a polynomial function. Finally, we perform illustrative numerical
examples to showcase the applicability and high accuracy of the proposed
interpolant, in the simple case of few points, as well as hundreds of points on
3D-grids using a spline-like interpolation, which compares favorably to
state-of-the-art spline interpolation methods